tppreqb - Metamath Proof Explorer

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Theorem tppreqb 4157

Description: An unordered triple is an unordered pair if and only if one of its elements is a proper class or is identical with one of the another elements. (Contributed by Alexander van der Vekens, 15-Jan-2018.)

**Assertion**
Ref	Expression
tppreqb	$/\$ $/\$

**Proof of Theorem tppreqb**
Step	Hyp	Ref	Expression
1		3ianor 988	. . . 4 $/\$ $/\$ $\/$ $\/$
2		df-3or 972	. . . 4 $\/$ $\/$ $\/$ $\/$
3	1, 2	bitri 249	. . 3 $/\$ $/\$ $\/$ $\/$
4		orass 522	. . . . 5 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
5		ianor 486	. . . . . . . 8 $/\$ $\/$
6		tpprceq3 4156	. . . . . . . 8 $/\$
7	5, 6	sylbir 213	. . . . . . 7 $\/$
8		tpcoma 4112	. . . . . . 7
9		prcom 4094	. . . . . . 7
10	7, 8, 9	3eqtr3g 2518	. . . . . 6 $\/$
11		orcom 385	. . . . . . . 8 $\/$ $\/$
12		ianor 486	. . . . . . . 8 $/\$ $\/$
13	11, 12	bitr4i 252	. . . . . . 7 $\/$ $/\$
14		tpprceq3 4156	. . . . . . 7 $/\$
15	13, 14	sylbi 195	. . . . . 6 $\/$
16	10, 15	jaoi 377	. . . . 5 $\/$ $\/$ $\/$
17	4, 16	sylbi 195	. . . 4 $\/$ $\/$ $\/$
18	17	orcs 392	. . 3 $\/$ $\/$
19	3, 18	sylbi 195	. 2 $/\$ $/\$
20		df-tp 4021	. . . 4
21	20	eqeq1i 2461	. . 3
22		ssequn2 3663	. . . 4
23		snssg 4149	. . . . . . 7
24		elpri 4036	. . . . . . . 8 $\/$
25		nne 2655	. . . . . . . . . 10
26		3mix2 1164	. . . . . . . . . 10 $\/$ $\/$
27	25, 26	sylbir 213	. . . . . . . . 9 $\/$ $\/$
28		nne 2655	. . . . . . . . . 10
29		3mix3 1165	. . . . . . . . . 10 $\/$ $\/$
30	28, 29	sylbir 213	. . . . . . . . 9 $\/$ $\/$
31	27, 30	jaoi 377	. . . . . . . 8 $\/$ $\/$ $\/$
32	24, 31	syl 16	. . . . . . 7 $\/$ $\/$
33	23, 32	syl6bir 229	. . . . . 6 $\/$ $\/$
34		3mix1 1163	. . . . . . 7 $\/$ $\/$
35	34	a1d 25	. . . . . 6 $\/$ $\/$
36	33, 35	pm2.61i 164	. . . . 5 $\/$ $\/$
37	36, 1	sylibr 212	. . . 4 $/\$ $/\$
38	22, 37	sylbir 213	. . 3 $/\$ $/\$
39	21, 38	sylbi 195	. 2 $/\$ $/\$
40	19, 39	impbii 188	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 184 $\/$ wo 366 $/\$ wa 367 $\/$ w3o 970 $/\$ w3a 971

wceq 1398

wcel 1823

wne 2649

cvv 3106

cun 3459

wss 3461

csn 4016

cpr 4018

ctp 4020

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1623 ax-4 1636 ax-5 1709 ax-6 1752 ax-7 1795 ax-10 1842 ax-11 1847 ax-12 1859 ax-13 2004 ax-ext 2432

This theorem depends on definitions: df-bi 185 df-or 368 df-an 369 df-3or 972 df-3an 973 df-tru 1401 df-ex 1618 df-nf 1622 df-sb 1745 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2651 df-v 3108 df-dif 3464 df-un 3466 df-in 3468 df-ss 3475 df-nul 3784 df-sn 4017 df-pr 4019 df-tp 4021

This theorem is referenced by: (None)

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